Ternary Egyptian fractions with prime denominator

TL;DR

This paper investigates the number of representations of fractions with prime denominators as sums of three unit fractions, providing improved bounds on their cumulative count over primes.

Contribution

It improves the upper bound on the sum of the counts of such representations for primes, refining previous estimates significantly.

Findings

Established a tighter upper bound: x (a0a0a0)^3 (a0a0)^2

Confirmed the asymptotic growth rate of the sum over primes

Enhanced understanding of Egyptian fractions with prime denominators

Abstract

For a prime number $p$, let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p) \ll x (\log x)^5$. We improve the upper bound, showing $\sum_{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$.